MEASURE, MODELING AND COMPENSATION OF FATIGUE-INDUCED DELAY DURING NEUROMUSCULAR ELECTRICAL STIMULATION

Transcription

1 MEASURE, MODELING AND COMPENSATION OF FATIGUE-INDUCED DELAY DURING NEUROMUSCULAR ELECTRICAL STIMULATION By FANNY BOUILLON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013

2 2013 Fanny Bouillon 2

3 To my grandparents 3

4 ACKNOWLEDGMENTS I would like sincerely to thank the persons who contributed to the success of this project: my advisor at the University of Florida, Dr. Warren Dixon, my advisor at Télécom Physique Strasbourg, Pr. Bernard Bayle, Ryan Downey for his precious help, advice and patience all over the year, Christophe Collet and Shelly Burleson, Lyn Straka and Abigail Nelson, the NCR August 2012/August 2013 group, Dr. Donald Bolser and Dr. Teresa Pitts. Thanks to my parents, sister and brothers for their support through this whole year. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF FIGURES LIST OF TABLES ABSTRACT CHAPTER 1 INTRODUCTION Context Literature Review Outline MODELING FATIGUE-BASED ELECTROMECHANICAL DELAY Equipment and Participants Protocol Results Mechanical Delays Force During Stimulation Gender Comparison Discussion Characterization of the EMD in Terms of NMES-Induced Fatigue First Model: Exponential Second Model: Sum of Exponentials Conclusion TIME-VARYING ELECTROMECHANICAL DELAY COMPENSATION IN NEU- ROMUSCULAR ELECTRICAL STIMULATION Muscle Stimulation Model Input Delay Model Control Design Previous Approaches Robust Integral of the Sign of the Error control technique (RISE) Control objective Stability analysis Experimental Results Material and Participants Protocol Results Discussion

6 4 CONCLUSION Achievements Future Work APPENDIX: PROOF OF P REFERENCES BIOGRAPHICAL SKETCH

7 Figure LIST OF FIGURES page 2-1 Equipment Position of the electrode pads for stimulation.photos courtesy of Kevin Wilt Definition of the measurement of the electromechanical delay Extract of the records of an experiment Evolution of the EMD Typical evolution of stimulation and force measurements Peak force under long stimulation Peak force under short stimulation Force during maximal voluntary contractions EMD function of force output during stimulation EMD function of force output during short pulse trains EMD function of the moment of the force peak during stimulation Exponential fitting curve and prediction bounds of the EMD in terms of the force production Fitting curve and prediction bounds of the EMD in terms of the force production Reference input for angular position tracking EMD for position tracking Closed-loop tracking without delay compensation Closed-loop tracking with time-varying delay compensation Comparison of both controllers Tracking error comparison between both controllers Evolution of time lags between reference and response

8 Table LIST OF TABLES page 2-1 Evolution of the EMD Evolution of the EMD Maximal force output values Maximal force output values Normalized values of the force decay rate under stimulation Voluntary contractions measurements Mean, median, maximal and minimal EMD of men and women Coefficients of determination following a single exponential curve fit Coefficients of determination following a sum of exponential curve fit Experimental data for both closed-loop tracking systems

9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MEASURE, MODELING AND COMPENSATION OF FATIGUE-INDUCED DELAY DURING NEUROMUSCULAR ELECTRICAL STIMULATION Chair: Warren E. Dixon Major: Mechanical Engineering By Fanny Bouillon August 2013 Neuromuscular Electrical Stimulation (NMES) is a developing method used for rehabilitation after surgery or for patients suffering from trauma and diseases. Functional Electrical Stimulation (FES) is the specific use of NMES to produce functional tasks, without the intervention of the nervous system. NMES training may be of long duration and NMES may induce muscle fatigue for the patients. The nonlinear response of the muscle is rendered more complex with external factors, including fatigue. Fatigue leads to a decreased muscle force in response to stimulation, and fatigue is suspected to lengthen the time lag from the onset of the electrical stimulus to the response of the muscle, termed electromechanical delay (EMD). Evolution of EMD was investigated on 5 male and 5 female subjects. The subjects muscles were fatigued through NMES exercises, consisting of constant voltage stimulation applied to the quadriceps femoris muscle group to produce isometric contractions. EMD was measured using monophasic electrical pulse trains of 0.5 seconds, characterized with constant frequency and voltage. Fatigue was induced with pulse trains of the same stimulation frequency and voltage applied over a longer period of time. Force production were measured throughout the trials for all the experiments. Fatigue was quantified by the decrease in force produced by the muscle when stimulated with constant parameters. The results showed an increase of the EMD with fatigue, up to 3 times its initial value. An exponential model was found to fit the evolution of the EMD in terms of fatigue. In order to track a desired 9

10 angular position trajectory, a time-varying input delay compensation was designed for nonlinear systems. The fatigue-varying EMD model was implemented in the controller as a time-varying input delay model. Experiments were conducted to determine the performance of the controller and the goodness of the EMD model. The experiments consisted of two sessions separated by several hours.the first session was used as a control test and did not involve any delay compensation. The controller used in the second session included time-varying input delay compensation adapted to NMES. The time-varying delay was compensated based on the estimation provided by the first session. The experiments run with this controller showed promising results, including a decreased delay between the input and the response. 10

11 CHAPTER 1 INTRODUCTION 1.1 Context Neuromuscular electrical stimulation (NMES) is the process of applying an electrical potential across a muscle to yield a muscle contraction. NMES is commonly used for rehabilitation after trauma, surgery, or to treat neuromuscular disorder and disease, such as spinal cord injury (SCI) and Parkinson s disease. NMES can be an alternative method to or complement traditional therapy. NMES can be used to produce functional tasks or movements where it is termed functional electrical stimulation (FES). Traditional NMES-based rehabilitation is to strengthen muscle, whereas more advanced methods focus on closed-loop control of the limbs. A challenge for these advanced methods is that skeletal muscle has a nonlinear response to an external electrical stimulus. The response depends on various factors related to the condition of the person who is stimulated, the stimulation parameters, and the muscle length and velocity. The onset of muscle fatigue is more rapid during (NMES) than during voluntary contractions [1 3]. The response to stimulation changes as the muscle fatigues [4 6], and thus, premature fatigue may limit the effectiveness of rehabilitation by limiting the duration of the performance of a task. The focus of this thesis is to investigate and compensate for changes in the electromechanical delay (EMD), defined here as the time difference between the onset of the stimulation and the onset of an actual movement of the stimulated limb, in the quadriceps femoris muscle group as a function of fatigue. 1.2 Literature Review Previous literature has related the dependency of the EMD with muscle type, or stimulation type, but the relationship between the delay and the fatigue state of the muscle during NMES is not clear. 11

12 Previous research targets the development of relationships between EMD and the intensity of contractions [7, 8], or the type of exercise (i.e. electrical stimulation versus voluntary contractions) [8]. EMD is also correlated to the maximal voluntary contraction and the torque produced by the muscle. Specifically, the results in [9], indicated that EMD is shorter for individuals who can produce strong maximal voluntary contractions. EMD variations in NMES has also been compared between different aged people [10, 11]. Both of these papers found a significant increased EMD in older subjects. EMD was also compared between men and women, but different results were observed: shorter EMD for men were reported in [12, 13], while the results of [11] indicated a difference of EMD between the two genders only among the year old range, and the results in [10] did not show any significant change for the 2 groups within their 30 subjects. EMD has been examined prior to and following a single fatiguing protocol during voluntary [14, 15] and NMES-induced [10] contractions. In 2001, Kubo found an actual decrease of the EMD with fatigue, after the subjects are trained [16]. Significant increases in EMD during voluntary fatiguing exercises were reported in [10, 15, 17, 18], where EMD was measured as the time lag between the onset of electrical activity (measured with electromyography) and the onset of force production. However, to the extent of the authors knowledge no study has examined or modeled the EMD with respect to NMES-induced fatigue. To provide an indicator of muscle fatigue during isometric contractions, this study will measure the force produced by the muscle in response to both electrical stimulation and voluntary contractions, as in [14, 19 21], through a force transducer. Few FES closed-loop systems were designed to include an input delay compensation. Studies in [22 25] involved PD controllers, without any EMD prediction term. Moreover, the results presented in those studies are not supported through mathematical proof of stability. The only prior work that accounts for EMD during NMES closed-loop is described in [26]. However, this result assumes a constant input delay. 12

13 1.3 Outline The following study aims to state the relationship between the EMD and the fatigue due to electrical stimulation of the quadriceps femoris muscle group. Fatigue experiments were executed while the time-varying EMD was measured. Chapter 1 describes the context and state of the art related to understanding EMD during NMES/FES. Only a few results examine the variations of EMD with fatigue. The only prior result that compensates for EMD during closed-loop control, assures the delay is constant. Therefore, this chapter highlights the novelty of the research described in this thesis. Chapter 2 first presents the methods and technological tools used to run NMES experiments and EMD collect data. This chapter also gathers the results of the evoked experiments and studies the evolution of the EMD with fatigue. The variations of the EMD are analyzed through different mechanical parameters, and a fatigue-varying model for the EMD in terms of force production during isometric contractions is investigated. Finally Chapter 3 applies a RISE-based NMES controller, that compensate for the time-varying EMD. The time-varying input delay is included in the RISE-based controller through a prediction term. The contributions of a time-varying input delay compensator are then compared with the same controller without the delay prediction. 13

14 CHAPTER 2 MODELING FATIGUE-BASED ELECTROMECHANICAL DELAY 2.1 Equipment and Participants The objective in this chapter is to correlate EMD with the fatigue state of the quadriceps femoris muscle group during isometric contractions. To study the evolution of the EMD, a group of ten volunteer subjects were stimulated to a level of fatigue while EMD was measured. The group was composed of five males and five females, aged 19 to 26. All participants signed an informed consent, approved by the Institutional Review Board of the University of Florida. Subjects were seated in a leg extension machine (LEM), depicted in Figure 2-1A). The leg extension machine was modified such that the subjects legs could be securely fixed to the machine. A boot was fixed to the leg extension machine through a force transducer that was used to record the force produced when the muscle was stimulated (see Figure 2-1B)). Stimulation was delivered through a pair of self-adhesive surface electrode pads, placed over the quadriceps femoris muscle group (see Figure 2-2). Pulse trains used in this study were to 30 Hz monophasic square electrical signals with a constant pulse width of 600 µs. The intersubject voltage amplitude varied from 20 to 40 V depending on each subject s response; however, the voltage amplitude remained constant for each subject. 2.2 Protocol The objective was to characterized the delayed response of the muscle in terms of fatigue, in healthy individuals. The initial EMD and force production for each subject were recorded from a resting position prior to experimental trials. To measure the EMD, a long stimulation duration is not necessary; thus, short pulse trains lasting 0.5 seconds were used to determine the moment when the muscle started to produce force. To obtain a more accurate measure of EMD, ten short pulse trains were applied, allowing a calculation of a mean value of the delay, immediately following a long stimulation. 14

15 The EMD is measured as the time difference between the onset of stimulation and the onset of muscle force production, where the threshold used to characterize the onset of force production was defined as 0.02% of the maximal voluntary contraction the subject performed at the beginning of the experiment (see Figure 2-3). Voluntary contractions were then performed to determine the maximal voluntary contraction (MVC) that the subject can achieve. These preliminary tests were designed such that they would not induce fatigue, and the corresponding measurements were used to normalize the subsequent EMD and force measurements for each subject. After pretrial tests were completed, trials were performed to fatigue the muscle. Fatigue trials consisted of long pulse trains with constant stimulation parameters lasting 20 to 35 seconds. Once calibrated for the subject, the duration remained constant for all the trials of a given subject. After each fatigue trial, another set of short pulse trains was applied to measure the average of the new state of the muscle in terms of the EMD and force production. Then the subject was asked to perform three maximal contractions. Finally, a longer set of short pulse trains was applied to measure of the evolution of the EMD during recovery. Each trial was repeated at least eight times for each subject. The number of cycles depended on subject comfort and level of fatigue. The duration of stimulation and the voltage applied to fatigue the muscle vary between subjects, but remain the same during all trials for a given subject. The duration varied among the subjects from twenty-five to thirty-five seconds. 2.3 Results The data gathered during the ten experiments are reported below Mechanical Delays Ten short pulse trains, each separated by one second, were applied to measure the average EMD after fatigue trials. Tables 2-1 and 2-2 report the values of the EMD obtained for each subject following each fatigue trial. Table 2-1 reports the raw values, 15

16 expressed in seconds. At the beginning of each trial, the EMD was recorded and used to normalize the subsequent values. These normalized values are reported in Table 2-2. The mean initial value of EMD was 46.8 ms, and ranged from 14 to 105 ms. For 20% of the experiment group, the variation of the EMD stayed between 1 and 1.5 of the initial value. Sixty percent of the subjects exhibited a maximal EMD which was 80% longer than the initial EMD. For 40% of the subjects the maximum is reached after three to eight long stimulation. The results clearly indicate that the EMD varies with fatigue, within a range, and in most cases, remains higher than its initial value until a recovery period. The next section will further investigate the change of EMD with fatigue Force During Stimulation An extract of a record is depicted in Figure 2-6, reporting the variations of the normalized force production during muscle fatigue for one subject. The horizontal cursors on the figure indicate the important decrease of the force, with the fatigue trials, whereas the level of the voluntary force values remains constant. The peak values of the force elicited from the long pulse trains are reported in Table 2-3 and are plotted in Figure 2-7. This data indicates that 30% of the subjects showed an increase of the force produced by their muscle under long stimulation, at the beginning of the fatiguing process. The initial value of the force decreased by 50% for 60% of the group at the end of the experiment, whereas 20% of the participants remained between 70% and 100% of their initial value. Under short stimulation, after the fatigue processes, the decrease is strengthened. In Table 2-4 and Figure 2-8, for 60% of the individuals, the peak force elicited from the short pulse trains declined by more than 50% after only 5 trials. While 20% of the subjects exhibited an increase in the peak force from the pretrial values, a decline of more than 60% of the peak force was measured following the final fatigue trial for 90% of subjects. 16

17 The instant of maximal force and the final value were used to measure the rate of the force decay under constant stimulation. The force decay rate is measured as the ratio of the difference between the highest normalized value of the force during each stimulation and the final normalized value of the force during the same trial, and the time difference between the moment when the force is the highest and the end of the trial r force decay = f max f final t final t fmax. The measurements are reported in Table 2-5. The force decay is the highest at the beginning, and then decreases, as the force peak decreases. However the variations are too disordered to develop a model Gender Comparison The set of volunteers was equally distributed between men and women. Table 2-7 segregates their respective mean, median, minimal and maximal values of the EMD, to compare the performance of each test group. As suggested by previous research [12, 13], there may be a difference in terms of EMD between men and women. The median, minimal and maximal EMD values for females are 36.9%, 25.2% and 63.2% lower than males, respectively. To determine if the two groups were statistically different, a Student t-test was performed with a significance level of The twosample assuming unequal variances test was realized with Excel for the maximal, minimal, mean, and median values of the EMD. The t-test was unable to reject the null hypothesis. Consequently, maximal, minimal, median or mean EMD for men subjects was not found to be statistically significantly different from the EMD of women subjects. This conclusion supports the results of [10] and [27] Discussion Most individuals described the fatigue sensation as a quick and strong increase that became constant. Our findings suggest that EMD varies significantly with NMESinduced fatigue and is in agreement with previous research studying voluntary [14, 15, 18] and NMES-induced contrctions [10]. There are variations in the experimental data in terms of the maximal normalized EMD and the time to reach the maximal relative 17

18 EMD. Fatigue was quantified through the measured force during stimulation. While EMD increased to an average value that was twice its initial value, the corresponding force produced by the muscle during stimulation was observed to decrease to less than half of the initial value, in both short and long pulse trains. Moreover, the protocol used did not allow the observation of any significant differences between men and women. 2.4 Characterization of the EMD in Terms of NMES-Induced Fatigue EMD is correlated to fatigue due to stimulation. The objective in this section is to build a fatigue versus EMD model for each individual. To represent the fatigue, a significant metric is necessary. Different a priori parameters are utilized in Figures 2-10 to 2-12, to determine the most appropriate metric to be used First Model: Exponential Based on the shape of the distribution of the data plotted in Figure 2-11, the following exponential function was developed to fit an approximate curve: τ = ae bx where τ is defined as the normalized EMD, x denotes the normalized force output, and a and b are unknown fitting coefficients. The exponential function fits the data with values of R 2 greater than 0.5 for 40% of the experiments, that is, the exponential model fits at least half of the variations of the data about the average for these individuals. When applied to the sum of all data, the R 2 value falls to (Figure 2-13). Even if less precise, the predictions bounds for the fitting include 94.12% of all the data points Second Model: Sum of Exponentials The following sum of exponential functions was also used to fit the data: (Figure (2-14)) τ = a 1 e b 1x + a 2 e b 2x (2 1) 18

19 where τ is defined as the normalized EMD, x denotes the normalized force output, and a 1,2 and b 1,2 are unknown fitting coefficients. With R2 values greater than 0.5 for 90% of the subjects, the exponential model explains more than half of the variation for these individuals. The model was also applied to the combined data set of all subjects resulting in more disparate behaviors among individuals. However, the R 2 value for individuals and for the global fitting were higher (Table 2-9),, reaching an R 2 value of The prediction bounds include 94.59% of the total number of data points Conclusion Our findings suggest that EMD varies significantly with NMES-induced fatigue and is in agreement with previous research with voluntary [14, 15, 18] and NMES-induced contractions [10]. The force production during isometric contractions was an effective metric to track fatigue during stimulation. Thus it was utilized to derive a model for EMD during fatigue. Individual R 2 for the second model were higher than those in the first model. The R 2 corresponding to a global fitting of all the subjects was lower than any individual R 2 but was still higher than in the case of the simple exponential. More than 36% of the variation for all subjects was explained, therefore this is a reasonable model for the evolution of EMD during NMES with respect to fatigue. The relationship between peak force and EMD may be better fitted by a non-exponential model, for small subsets of subjects. Additionally, the reasons for this potential variance among a population of human subjects are not clear at this time. The model was defined in terms of discrete measures of the force production, for a given voltage value. Because a value of the EMD is needed continuously over the stimulation, (2 1) can be expressed as: τ = a 1 e b 1 f N/v N + a2 e b 2 f N/v N (2 2) where f N R + denotes the normalized force output, expressed as the ratio of the actual force over the initial force value f /f 0. v N R + is the normalized voltage value used 19

20 to measure EMD, and thus can be written as v /v 0, where v is the positive and nonzero actual voltage and v 0 is the positive constant voltage used to determine the initial EMD. Based on these definitions, a measure of the EMD can be expressed continuously while the voltage is varying, with the measure of the corresponding force output. As a result, the actual model of the fatigue-varying EMD for each individual can be obtained with the initial value of the EMD denoted as τ 0 and measured during pre-trial tests: τ = τ 0 ( a1 e b 1p + a 2 e b 2p ) (2 3) where τ R + denotes the current EMD, and τ 0 the initial EMD. The ratio f /f 0 v/v 0 is denoted by p R +, where v 0 R +, and f 0 R + denote respectively the initial voltage input and force output, from a resting situation. Thus v R + and f R + correspond to the voltage of the stimulation, and corresponding force produced by the muscle. Finally, a 1, a 2, b 1, and b 2 are fitting coefficients, with a 1, a 2 > 0 and b 1, b 2 < 0. Based on the expression in (2 3), the EMD is bounded 0 < τ τ 0 ( a1 e b 1p min + a 2 e b 2p min ) = ϕ1 (2 4) where p min R is the minimum value achieved by p and ϕ 1 R + is constant. Provided that f, f and v, v are bounded, the rate of change of the EMD can also be bounded τ ϕ 2 (2 5) where ϕ 2 R + is constant. As EMD may affect FES task performance, the model is developed as a potential aid for designing NMES methods that account for EMD [26]. 20

21 A) Leg extension machine B) Force transducer Figure 2-1. A) Testbed used for the experiment: a leg extension machine, legs are decoupled, and rigid boots are set up to maintain the subject s leg. Adjustments are possible: moving the back of seat forward and backward, as well as moving the boot up and down, depending on the subject s height. B) A force transducer is fixed to the leg extension machine, to measure the force produced by the leg whether the muscle is under stimulation or not. Photos courtesy of Fanny Bouillon. Figure 2-2. Position of the electrode pads for stimulation.photos courtesy of Kevin Wilt. 21

22 Figure 2-3. Definition of the measurement of the electromechanical delay: time lag between the onset of stimulation (cursor 1) and the first movement of the limb (cursor 2). The cursor 2 is set so that the force value corresponds to 0.2% of the maximal voluntary contraction of the subject. Figure 2-4. Extract of the records of an experiment. Two cycles are shown: the top graph is the non amplified signal of the stimulation, measured in volts. The bottom graph is the signal from the force transducer measuring the force output, normalized with the initial value of the MVC. First, a long train of pulses is sent to stimulate the muscle. Then short pulse trains are applied to measure the EMD. Finally the subject performs voluntary contractions without stimulation. 22

23 Normalized EMD Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 Subject 6 Subject 7 Subject 8 Subject 9 Subject Experiment A) Individual curves of evolution of the delay for each individual B) Evolution of the distribution of the values of the EMD among the ten subjects. Figure 2-5. Representation of the EMD throughout the experiment for the ten subjects. A) Individual curves of evolution of the delay for each individual. B) Evolution of the distribution of the values of the EMD among the ten subjects. 23

24 A) First fifty seconds of an experiment. B) Next fifty seconds of the same experiment. Figure 2-6. Extract of a 100-second experiment, showing the evolution of the normalized force production measurements, while the subject is being stimulated and performs voluntary exercises. A) Zero to 50s. B) Fifty to 100s. The top plots in A) and B) correspond to the stimulation voltage, received by the subjects. The bottom plots in A) and B) represent the force measurements. The horizontal lines are on the same levels on A) and B), corresponding to the average level of the initial forces for both stimulation and voluntary contractions, and the final value reached during stimulation. 24

25 Figure 2-7. Evolution of the highest normalized values of the force produced under long stimulation for the ten experiments. Figure 2-8. Evolution of the normalized values of the force produced under brief stimulation for the ten experiments. Figure 2-9. Evolution of the maximal voluntary contraction the 10 subjects can perform respectively. 25

26 Figure EMD versus force peak during long stimulation for the 10 subjects. EMD and force measurements are normalized with their respective values recorded during pretrial test, when the muscle was in a resting position. Figure EMD versus force output during short pulse trains for the 10 subjects. Figure EMD versus instant of peak for the 10 subjects. The instant of peak corresponds to the time it took for the force to reach its highest value during a long pulse train. 26

27 4.5 4 Data points Exponential fitting curve Pred bnd Normalized EMD Normalized force production during short stimulation Figure Representation of the EMD in terms of the force output produced by the leg. Each point represents the values recorded for all the experiments. The whole set of points can be approximated by an exponential function, represented by the dashed line Data points Double exp fitting curve Pred bnds Normalized EMD Normalized force production during short pulse trains Figure Global fitting curve, and respective prediction bounds, of the EMD in terms of the force during short stimulation. The main dashed line represents the double exponential approximation (R 2 =0.3637). 27

28 Table 2-1. Evolution of the EMD raw values throughout the fatiguing experiment of each of the ten healthy subjects, expressed in seconds. The set number corresponds to the number of long stimulation the subject received, the corresponding EMD is measured immediately following stimulation. The number of fatigue sets depended on the comfort of each subject. Subject > initial value set set set set set set set set set set set set set set maximal delay number of sets to reach maximal delay

29 Table 2-2. Evolution of the EMD throughout the fatiguing experiment of each of the ten healthy subjects. The initial value is set to 1 to normalized each experiment, as absolute values were not to be compared from one subject to another. The set number corresponds to the number of long stimulation the subject received, the corresponding EMD is measured right after it. The number of fatigue sets depended on the pain and comfort situation of each subject. Subject> Mean Standard deviation set set set set set set set set set set set set set set maximal delay number of sets to reach maximal delay

30 Table 2-3. Measurements of the force produced by the leg: maximum reached during stimulation. Each subject s raw data is normalized in terms of their respective initial value (set to 1). Subject > Mean Standard deviation Table 2-4. Measurements of the force output under short pulses (half a second) right after a fatiguing stimulation. Each subject s raw data is normalized in terms of their respective initial value (set to 1). Subject > Mean Standard deviation

31 Table 2-5. Values of the force decay rate under stimulation, expressed in s 1 : measured as the speed at which the normalized force is decreasing after having reached its maximum of the stimulation. Subject > Mean Standard deviation Table 2-6. Voluntary contractions measurements. The force produced by each of the subjects when they perform maximal voluntary contractions is reported in this table. Each value is normalized with the initial performance of the corresponding subject. Subject > Mean Standard deviation

32 Table 2-7. Segregation of the mean, median, maximal and minimal values of the EMD, expressed in seconds, of men and women. The values reported here were used to test the difference between the two groups through a Student t-test., and the result of the test for each parameter is reported in the last row of the table. Mean EMD Median EMD Maximum EMD Minimum EMD Men Women P Table 2-8. Coefficients of determination following an exponential curve fit of the form τ = ae bx Subject # R 2 Subject # R ALL Table 2-9. The coefficient of determination following a curve fit of the form τ = a 1 e b 1x + a 2 e b 2x. Subject # R 2 Subject # R ALL

33 CHAPTER 3 TIME-VARYING ELECTROMECHANICAL DELAY COMPENSATION IN NEUROMUSCULAR ELECTRICAL STIMULATION The motivation to improve the NMES rehabilitation comfort and performances leads to the objective of adapting the stimulation intensity applied to the current state of the muscle. This chapter adapts a controller, designed to compensate for time-varying input delay in nonlinear systems, to FES. 3.1 Muscle Stimulation Model The musculoskeletal dynamics with one degree of freedom corresponding to the rotation about the knee joint, is defined as [28] M e + M g + M v + M I + d = T (t τ (t)) (3 1) where M I ( q) R corresponds to the inertia of the shank-foot system, M e (q) R denotes the elastic effects due to joint stiffness, M g (q) R denotes the gravitational component, and M v ( q) R the viscous effect due to damping in the musculotendon complex, d (t) R represents unknown bounded disturbances, and T (t τ (t)) R corresponds to the torque produced at the knee joint by the electrical stimulation. The generalized states are defined as q (t), q (t), q (t) R n and correspond to the angular position, velocity and acceleration of the lower limb. The inertia component is defined as M I ( q) = J q where J R denotes the unknown inertia of the shank-foot system. The gravitational effects can be modeled as M g (q) = mgl sin (q) where m, g, l R respectively are the unknown mass of the system, the unknown distance between the knee joint and the mass center, and the gravitational acceleration. The elastic effects are defined as M e (q) = k 1 e k2q (q k 3 ) where k 1, k 2, k 3 R are unknown positive constants. Finally, the viscous effects are defined as M v = B 1 tanh ( B 2 q) + B 3 q where B 1, B 2, B 3 R are unknown positive constants. The knee torque is related to the muscle tendon force F (q, q, t) through T = ζf where ζ (q) is a positive unknown nonlinear moment arm. The muscle force generated at 33

34 the tendon is expressed as F = ΓV where V (t) is the voltage applied and Γ (q, q) is an unknown nonlinear function. Assumption1 The moment arm is assumed to be a non-zero, positive and bounded function whose first and second time derivatives exist. The function Γ is assumed to be non zero, positive and bounded function, whose first and second time derivatives exist and are bounded. Assumption2 The first and second time derivatives of an auxiliary non-zero, unknown scalar function Ω (q, q), defined as Ω = ζγ, are assumed to exist and be bounded. Assumption3 The unknown disturbance d (t) is bounded, as well as its first and second time derivatives. Equation (3 1) can be rewritten as q = f (q, q, t) + d + u (t τ (t)) (3 2) where f (q, q, t) = 1 J ( mgl sin (q) k1 e k 2q (q k 3 ) B 1 tanh ( B 2 q) B 3 q ) (3 3) is a nonlinear unknown C 2 function, uniformly bounded in t, d corresponds to smooth disturbances, u denotes the delayed control input, and τ the positive time-varying input delay. Assumption4 The nonlinear disturbance term and its first two time derivatives exist and are bounded by known constants. Assumption5 The desired trajectory q d R is designed such that q (i), i [0, 4] exist and are bounded by known positive constants, where q (i) d derivative of q d. d denotes the i th time Assumption6 The input delay is bounded such that 0 τ ϕ 1 and the rate of change of the delay is bounded such that τ ϕ 2 < 1 where ϕ 1,2 are known constants, and ϕ 1 + ϕ 2 < 1. 34

35 3.2 Input Delay Model The fatigue-varying electromechanical delay model (2 1) that was determined in Chapter 3 was ultimately to be utilized for FES. However, the model was developed under isometric conditions and requires force measurements. Experiments allow the measure of the force, but the resulting output signal has a large signal to noise ratio without the use of a low-pass filter. Thus additional lag would be added to the force signal, interfering with the EMD compensation. Moreover, position tracking involves the movement of the lower limb. A new set up of the leg extension machine has to be done, to give the leg its rotational degree of freedom about the knee. However, the value that will be measured will not correspond to the muscle production force. To retrieve this quantity, and use the model in the previous chapter, additional modeling of the muscle, and sensors would be necessary. Instead of adding sensors, the input delay will be modeled as a function of time. To do so, electrical pulses are successively applied to the lower limb, and the EMD is measured as the time lag between the onset of stimulation and the onset of the movement of the leg. This latter is defined as 0.02% of the maximal reference angle. Then an expression, involving time, is derived to fit the evolution of the EMD. This expression corresponds to the time-varying input delay model, and is specific to a reference trajectory. The measure of the EMD and its modeling will be detailed in the Experimental Results Section. 3.3 Control Design Previous Approaches Linear systems are well controlled in various ways in the literature [29 34], including constant input delay issues for linear systems [35 37]. These results were also extended later for time-varying input delays [38 44]. Nonlinear systems have also been studied, whether they involve constant [45 52] or time-varying [45, 53 62] state delays, but input delays in uncertain systems have received less consideration. However 35

36 backstepping and robust techniques were used [61, 63 65], as well as predictor-based techniques [37, 64, 65], but required no disturbances. For this study, the controller in [61] was utilized and adapted to neuromuscular electrical stimulation. To compensate for the input delay, a predictor-like error signal based on previous control values is used to yield a delay-free open-loop system, allowing control design flexibility. A Lyapunov stability analysis then demonstrates the achievement of semi-global asymptotic tracking in the presence of model uncertainty, smooth disturbances and time-varying input delay Robust Integral of the Sign of the Error control technique (RISE) Control objective To quantify the control objective, the tracking error e 1 R is defined as e 1 = q d q (3 4) Moreover, two auxiliary tracking errors e 2 R and r R are defined as e 2 = e 1 + α 1 e 1 (3 5) r = e 2 + α 2 e 2 + e u (3 6) where α 1, α 2 are positive real constant control gains, and e u R is the difference between the delayed control input and the actual control input defined as e u = u (t τ (t)) u (t) (3 7) Furthermore, an auxiliary filter e uf R is defined as the solution to the differential equation e uf = α 2 e uf + e u (3 8) 36

37 Substituting (3 8) into (3 6) allows (3 6) to be expressed as r = η + α 2 η (3 9) where η R is defined as η = e 2 + e uf (3 10) Then substituting for η into r gives r = q d f ( q, q, t ) d u (t τ) + α 1 e 1 + e uf + α 2 η. (3 11) To obtain the open-loop tracking error, (3 8), (3 7) and (3 10) are substituted in (3 11), and then some algebraic manipulation f (q d, q d, t), leads to r = S 1 + S 2 u (3 12) where S 1, S 2 R are defined as S 1 = f (q d, q d, t) f (q, q, t) + α 1 e 1 + α 2 e 2 (3 13) S 2 = q d f (q d, q d, t) d. (3 14) The use of (3 7) eliminates the delayed terms. Based on the open-loop dynamics in (3 12), the controller is designed as u = (k s + 1) (e 2 e 2 (t 0 )) + ν (3 15) where v is the Filippov solution to ν = (k s + 1) (α 2 e 2 + e u ) + βsgn (η) (3 16) and k s and β are positive constant control gains. The existence of Filippov solutions can be established for v K [h 1 ] (e 2, e u, η), where h 1 R is defined as the right-hand side of (3 16), and K [h 1 ] coh 1 (e 2, e u, B (η, δ) S m ), where δ R, denotes δ>0 µ(s m)=0 µ(s m)=0 37

38 the intersection over sets S m of Lebesgue measure zero, co denotes convex closure, and B (η, δ) = {ς R n η ς < δ} [66, 67]. Thus the closed-loop error system can be written ṙ = Ñ + N d e 2 (k s + 1) r βsgn (η) (3 17) with Ñ = S 1 + e 2 and N d = S 2. (3 18) The following inequalities can be developed from the expression in (3 18) and the Mean Value Theorem: Ñ ρ ( z ) z (3 19) where z = [ ] e T 1, e T 2, r T, e T T u (3 20) and ρ is a positive, non decreasing and invertible function. The expression in (3 18) can also be upper-bounded as N d ζ Nd1 N d ζ Nd2 (3 21) where ζ Nd1, ζ Nd2 are known positive constants, and let ź R 3 be defined as ź [ e T 1, e T 2, r T ] T. (3 22) To facilitate the subsequent analysis, an auxiliary constant σ is defined as { σ = min α 1 1 2, α 2 1, 4k s 3 ωϕ 1 (k s + 1) 2, ω (1 ϕ } 1 ϕ 2 ), (3 23) ϕ 1 where ω is a known adjustable positive constant. 38

39 Stability analysis Theorem 1. The controller in (3 15) and (3 16) ensures uniformly ultimately bounded stability in the sense that e 1 (t) ɛ 0 exp ( ɛ 1 t) + ɛ 2, (3 24) where ɛ 0, ɛ 1, ɛ 2 R are positive constants, and provided the control gains satisfy the following conditions: α 1 > 1 2 (3 25) { α 2 > 1 (3 26) β > ζ Nd1 + ζ N d2 α 2 (3 27) 4σk s > 3ρ 2 ( z (t 0 ) ) (3 28) ϕ 1 ω > 2 (1 ϕ 1 ϕ 2 ) (3 29) k s (k s + 1) 2 > 3ωϕ 1. (3 30) ( PROOF. Let D y R 5 ý < ρ 1 4σks, where ý [ ] y T, e T T u R 6, be 3 an open and connected set containing y = 0, where y R 5 is defined as )} [ ] T y ź T P Q. (3 31) In (3 31), the auxiliary function P R is defined as a Filippov solution to the following differential equation P = r T (N d βsgn (η)), n P (t 0 ) = β η i (t 0 ) η i (t 0 ) T N d (t 0 ) (3 32) i=1 where the subscript i = 1, 2,..., n denotes the i th element of a vector. Similar to the development in (3 16), existence of solutions P for (3 32) can be established. Provided 39

40 the sufficient condition for β in (3 27) is satisfied, P (t) 0, t [0, ) (See the Appendix A for details). Additionally, let Q R denote an LK functional, defined as where ω was introduced in (3 23). ˆ t (ˆ t ) Q ω u (θ) 2 dθ ds (3 33) t τ(t) s Let V : D R be a continuously differentiable function defined as V 1 2 et 1 e et 2 e rt r + P + Q (3 34) which satisfies the following inequalities: λ 1 y 2 V (y) λ 2 y 2 (3 35) where λ 1, λ 2 R + are positive constants. Consider a set { ( )} B σ y D y < ρ 1 4σks D (3 36) 3 and let S D B σ be defined as { S D y B σ y < λ1 λ 2 ρ 1 ( )} 4σks. (3 37) 3 Let y be a Filippov solution to the closed-loop system ẏ = h 3 (y, t) such that y (t 0 ) S D, where h 3 : R 6 [0, ) R 6 denotes the right-hand side of the closed-loop error system. The time derivative of (3 34) exists almost everywhere (a.e.), i.e., for almost all t [t 0, t f ], and V (y (t)) a.e. Ṽ (y (t)) where Ṽ = [ ] T ξ V L (y) ξt K ė T 1, ė T 2, ṙ T, P 1 2 P, 1, Q 2 Q 2 2 and V is the generalized gradient of V [68]. Throughout the subsequent discussion, let a.e. refer to almost all t [0, ). Since V is continuously differentiable, where V [ Ṽ V T K ė T 1, ė T 2, ṙ T, P 1 2 P 2 ] T, Q 1 2 Q (3 38) 2 [ e T 1, e T 2, r T, 2P 1 2, 2Q 1 2 ] T. Using the calculus for K from [67], applying the Leibniz Rule to determine the time derivative of (3 33), and substituting (3 4)-(3 6), 40